Piecewise-linear (PWL) canard dynamics : Simplifying singular perturbation theory in the canard regime using piecewise-linear systems

Abstract : In this chapter we gather recent results on piecewise-linear (PWL) slow-fast dynamical systems in the canard regime. By focusing on minimal systems in $\mathbb{R}^2$ (one slow and one fast variables) and $\mathbb{R}^3$ (two slow and one fast variables), we prove the existence of (maximal) canard solutions and show that the main salient features from smooth systems is preserved. We also highlight how the PWL setup carries a level of simplification of singular perturbation theory in the canard regime, which makes it more amenable to present it to various audiences at an introductory level. Finally, we present a PWL version of Fenichel theorems about slow manifolds, which are valid in the normally hyperbolic regime and in any dimension, which also offers a simplified framework for such persistence results.
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Submitted on : Wednesday, December 6, 2017 - 2:20:08 PM
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Mathieu Desroches, Soledad Fernández-García, Martin Krupa, Rafel Prohens, Antonio Teruel. Piecewise-linear (PWL) canard dynamics : Simplifying singular perturbation theory in the canard regime using piecewise-linear systems. Nonlinear Systems, 1, ⟨Springer⟩, In press, Mathematical Theory and Computational Methods, 978-3-319-66765-2. ⟨10.1007/978-3-319-66766-9_3⟩. ⟨hal-01651907⟩

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